Problem: There exists a constant $c,$ so that among all chords $\overline{AB}$ of the parabola $y = x^2$ passing through $C = (0,c),$
\[t = \frac{1}{AC} + \frac{1}{BC}\]is a fixed constant.  Find the constant $t.$

[asy]
unitsize(1 cm);

real parab (real x) {
  return(x^2);
}

pair A, B, C;

A = (1.7,parab(1.7));
B = (-1,parab(-1));
C = extension(A,B,(0,0),(0,1));

draw(graph(parab,-2,2));
draw(A--B);
draw((0,0)--(0,4));

dot("$A$", A, E);
dot("$B$", B, SW);
dot("$(0,c)$", C, NW);
[/asy]
Answer: To get a handle on the constant $t,$ we can look at some particular cases.

Suppose we let $AB$ approach a vertical line.  Then $\frac{1}{AC}$ approaches 0, and $B$ approaches $(0,0),$ so $\frac{1}{AC} + \frac{1}{BC}$ approaches $c.$  Hence,
\[t = \frac{1}{c}.\]Now, suppose we take $A = (\sqrt{c},c)$ and $B = (-\sqrt{c},c).$  Then
\[t = \frac{1}{AC} + \frac{1}{BC} = \frac{1}{\sqrt{c}} + \frac{1}{\sqrt{c}} = \frac{2}{\sqrt{c}}.\]Hence, $\frac{1}{c} = \frac{2}{\sqrt{c}},$ so $\sqrt{c} = \frac{1}{2},$ and $c = \frac{1}{4}.$  Therefore, $t = \boxed{4}.$  (Note that this makes $C$ the focus of the parabola.)

For a full solution, let's check that this value works.  Let $y = mx + \frac{1}{4}$ be the equation of line $AB.$  Setting $y = x^2,$ we get
\[x^2 = mx + \frac{1}{4},\]or $x^2 - mx - c = 0.$  Let $x_1$ and $x_2$ be the roots of this equation.  By Vieta's formulas, $x_1 + x_2 = m$ and $x_1 x_2 = -\frac{1}{4}.$

Also, $A$ and $B$ are $(x_1,x_1^2)$ and $(x_2,x_2^2)$ in some order, so
\begin{align*}
\frac{1}{AC} + \frac{1}{BC} &= \frac{1}{\sqrt{x_1^2 + (x_1^2 - \frac{1}{4})^2}} + \frac{1}{\sqrt{x_2^2 + (x_2^2 - \frac{1}{4})^2}} \\
&= \frac{1}{\sqrt{x_1^2 + x_1^4 - \frac{1}{2} x_1^2 + \frac{1}{16}}} + \frac{1}{\sqrt{x_2^2 + x_2^4 - \frac{1}{2} x_2^2 + \frac{1}{16}}} \\
&= \frac{1}{\sqrt{x_1^4 + \frac{1}{2} x_1^2 + \frac{1}{16}}} + \frac{1}{\sqrt{x_2^4 + \frac{1}{2} x_2^2 + \frac{1}{16}}} \\
&= \frac{1}{\sqrt{(x_1^2 + \frac{1}{4})^2}} + \frac{1}{\sqrt{(x_2^2 + \frac{1}{4})^2}} \\
&= \frac{1}{x_1^2 + \frac{1}{4}} + \frac{1}{x_2^2 + \frac{1}{4}}.
\end{align*}We have that $x_1^2 x_2^2 = (x_1 x_2)^2 = \left( -\frac{1}{4} \right)^2 = \frac{1}{16}$ and
\[x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2 = m^2 + \frac{1}{2}.\]Hence,
\begin{align*}
\frac{1}{x_1^2 + \frac{1}{4}} + \frac{1}{x_2^2 + \frac{1}{4}} &= \frac{x_1^2 + \frac{1}{4} + x_2^2 + \frac{1}{4}}{(x_1^2 + \frac{1}{4})(x_2^2 + \frac{1}{4})} \\
&= \frac{x_1^2 + x_2^2 + \frac{1}{2}}{x_1^2 x_2^2 + \frac{1}{4} (x_1^2 + x_2^2) + \frac{1}{16}} \\
&= \frac{m^2 + 1}{\frac{1}{16} + \frac{1}{4} (m^2 + \frac{1}{2}) + \frac{1}{16}} \\
&= \frac{m^2 + 1}{\frac{1}{4} m^2 + \frac{1}{4}} \\
&= 4.
\end{align*}